Gukov on categorification and gauge theory II

Since I went into quite a bit of detail about Saturday’s Gukov talks, I feel a bit obligated to discuss the second half, at least a little.

This talk was mostly restricted to the so-called “GL-twist” (this is Gukov’s name. I know it’s terribly confusing to use GL as an abbreviation for geometric Langlands, but I don’t know any better name), with is a 4-d TQFT, coming from one with N=4 supersymmetry, which localizes on the space of flat connections for the complexified gauge group (thought of as the space of Higgs bundles).

So, if one is a quantum topologist looking at this gauge theory picture, then getting an invariant of 4-manifolds with boundary is nice, but what one really wants is to put knots into the boundary of this 4-manifold, add a surface between them, and to get something associated to that.

Specifically, a vector space for each 3-manifold with a knot, and a morphism between these vector spaces for each surface between them. To do this we need “observables” attached to surfaces, analogous to the Wilson loops used to define the quantum invariants in \mathrm{SU}(n) Chern-Simons theory.

The definition of these “surface operators” is more like the `t Hooft operators in 3-d. To apply such an operator, one should restrict one’s path integral to a subset of connections satisfying a certain ansatz on the transverse directions (basically one perturbes Hitchin’s equations on the transverse surface to your own and takes a hyperkähler moment map level different from 0, but with all the curvature concentrated at the intersection point with your surface), and should modify the path integral a bit using a B-field on your surface.

Unfortunately, he didn’t go into much more detail after this, and instead switched to talking about getting affine braid group actions on derived categories of coherent sheaves on these moduli spaces of solutions to the deformed Hitchin’s equations (but of course, in a very physicsy, non-specific way), basically by finding a subset of the parameter space one can run around in with changing the derived category (which is very confusing, because apparently his category was depending on some stability condition based on the B-field which he would not write down or explain) whose fundamental group is given by the affine braid group, and so…voilà!

By the time we had finished this, and everyone had asked their questions, we had been going for about 3 and half hours (after 3 hours the day before), and it was definitely quitting time.

Except that people spent a good half hour asking Sergei questions after the official question period, particular about what actual knot invariants one will get out of these TQFTs. The answer is, sadly, “counting type invariants” such as the Casson invariant, signature and Alexander polynomial. Apparently to get quantum invariants, one has to allow “coisotropic branes.” Oh well. I suppose there will always be more gauge theory to learn.


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